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Mirrors > Home > MPE Home > Th. List > fnconstg | Structured version Visualization version GIF version |
Description: A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.) |
Ref | Expression |
---|---|
fnconstg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstg 6005 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | |
2 | ffn 5958 | . 2 ⊢ ((𝐴 × {𝐵}):𝐴⟶{𝐵} → (𝐴 × {𝐵}) Fn 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 {csn 4125 × cxp 5036 Fn wfn 5799 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: fconst2g 6373 ofc1 6818 ofc2 6819 caofid0l 6823 caofid0r 6824 caofid1 6825 caofid2 6826 fnsuppres 7209 fczsupp0 7211 fczfsuppd 8176 brwdom2 8361 cantnf0 8455 ofnegsub 10895 ofsubge0 10896 pwsplusgval 15973 pwsmulrval 15974 pwsvscafval 15977 xpsc0 16043 xpsc1 16044 pwsco1mhm 17193 dprdsubg 18246 pwsmgp 18441 pwssplit1 18880 frlmpwsfi 19915 frlmbas 19918 frlmvscaval 19929 islindf4 19996 tmdgsum2 21710 0plef 23245 0pledm 23246 itg1ge0 23259 mbfi1fseqlem5 23292 xrge0f 23304 itg2ge0 23308 itg2addlem 23331 bddibl 23412 dvidlem 23485 rolle 23557 dveq0 23567 dv11cn 23568 tdeglem4 23624 mdeg0 23634 fta1blem 23732 qaa 23882 basellem9 24615 ofcc 29495 ofcof 29496 eulerpartlemt 29760 matunitlindflem1 32575 matunitlindflem2 32576 ptrecube 32579 poimirlem1 32580 poimirlem2 32581 poimirlem3 32582 poimirlem4 32583 poimirlem5 32584 poimirlem6 32585 poimirlem7 32586 poimirlem10 32589 poimirlem11 32590 poimirlem12 32591 poimirlem16 32595 poimirlem17 32596 poimirlem19 32598 poimirlem20 32599 poimirlem22 32601 poimirlem23 32602 poimirlem28 32607 poimirlem29 32608 poimirlem31 32610 poimirlem32 32611 broucube 32613 cnpwstotbnd 32766 eqlkr2 33405 pwssplit4 36677 mpaaeu 36739 rngunsnply 36762 ofdivrec 37547 dvconstbi 37555 zlmodzxzscm 41928 aacllem 42356 |
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